Trigonometric Functions
Sine, Cosine, Tangent (Review)
• For a RIGHT TRIANGLE
– Sine – Opposite over Hypotenuse : sin θ
– Cosine – Adjacent over Hypotenuse : cos θ
– Tangent – Opposite over Adjacent : tan θ
• SOH
• CAH
• TOA
Example: Finding Trigonometric Ratios
Find the value of the sine, cosine, and
tangent functions for θ.
sin θ =
cos θ =
tan θ =
Special Right Triangles
Algebra 2 – 10.1
• 30/60/90 – side relationships 1: 3 :2
• 45/45/90 – side relationships 1 : 1 : 2
Example: Finding Side Lengths of 30 – 60 – 90 Triangle
Use a trigonometric function to find the value of x.
The sine function relates the
opposite leg and the
hypotenuse.
°
Substitute 30° for θ, x for opp,
and 74 for hyp.
Substitute
x = 37
for sin 30°.
Multiply both sides by 74 to solve for x.
Example: Finding Side Lengths of 45 – 45 – 90 Triangle
Use a trigonometric function to find the value of x.
The sine function relates the
opposite leg and the
hypotenuse.
°
° x for opp, and
Substitute 45 for θ,
20 for hyp.
Substitute
for sin 45°.
Multiply both sides by 20 to solve for x.
Angles of Rotation
Algebra 2 – 10.2
• Standard Position
– Vertex is origin
– One ray is positive x axis
• Initial Side – x axis ray
• Terminal Side – other ray
• Angle of Rotation
– Maintain initial side and rotate to terminal side
Example: Drawing Angles in Standard Position
Draw an angle with the given measure in standard position.
A. 320°
Rotate the
terminal side 320°
counterclockwise.
B. –110°
Rotate the
terminal side –110°
clockwise.
C. 990°
Rotate the
terminal side 990°
counterclockwise.
Reference Angle
• Positive acute angle of the triangle
• Quadrant of Reference Angle determines sign
of functions
Example: Finding Reference Angles
Find the measure of the reference angle for each given angle.
A.  = 135°
B.  = –105°
–105°
The measure of the reference angle is 45°.
The measure of the reference angle is 75°.
Reference Angle
To determine the value of the
trigonometric functions for an
angle θ in standard position, begin
by selecting a point P with
coordinates (x, y) on the terminal
side of the angle. The distance r
from point P to the origin is given
by
.
Trig to Circles
Algebra 2 – 10.2
• If vertex is (0,0) - trig uses x and y coordinates
of point
– Radius (r) is
– Sine is y/r, Cosine is x/r, and Tangent is y/x
Example: Finding Values of Trigonometric Functions
P (–6, 8) is a point on the terminal side of  in standard
position. Find the exact value of the three trigonometric
functions for θ.
Step 1 Plot point P, and use it to sketch a right
triangle and angle θ in standard position. Find r.
r = (-6) +(8) = 36 + 64 = 100 =10
2
sin θ=
2
cos θ=
tan θ=
Examples
• Use the following coordinates to determine
the trigonometric functions (sin, cos, tan):
1.
2.
3.
4.
(3, 4)
(-3, 4)
(-3, -4)
(3, -4)
Signs in Quadrants
• The location of the reference angle
determines the sign of the functions
Radians and Degrees
Algebra 2 – 10.3
• Radian – Angle measure based on arc length
– Circumference of circle = 2πr
– Complete revolution of circle = 360o
• Relationship of radians to degrees is 2π = 3600
Unit Circle
Algebra 2 – 10.3
• Circle with a radius of 1
• Relation of radians, degrees and the sine and
cosine of the related angles
• Coordinates of point on circle are (cosθ, sinθ)
– Cosine is the x coordinate
– Sine is the y coordinate
• Use 30-60-90 & 45-45-90 triangles
Building Unit Circle
Unit Circle
Graphing Sin/Cos Functions
Algebra 2 – 11.1
• Periodic – repeats exactly at a given interval
– Intervals are called cycles
– Length of the cycle is the period
• Sin & Cos are Periodic
– Values are the y & x
values on unit circle
– Period is 2π • 1 complete rotation
Periodic vs NOT Periodic
• To Determine Period
– Peak to Peak or Trough to Trough
Example: Identifying Periodic Functions
Identify whether the function is periodic.
If the function is periodic, give the period.
The pattern repeats exactly, so the
function is periodic. Identify the period by
using the start and finish of one cycle. This
function is periodic with a period of .
Sin & Cos
• Sin & Cos are Periodic
– Values are the y & x values on unit circle
– Period is 2π - 1 complete rotation
Amplitude
• The amplitude of sine and cosine functions is
half of the difference between the maximum
and minimum values of the function. The
amplitude is always positive.
• This will be the difference from neautral (or 0)
to the peak or trough or ½ the distance from
the peak to trough
Transformations
• Changing the Amplitude and Period
– Amplitude is indicated by the a value (in front)
– Period is the b value in front of the x (2π/b)
To Graph Sine/Cosine Functions
• Determine the Peak & Trough Value
• Determine the Period
– Peak-peak/trough-trough
• Set up a table – 5 columns – 2 rows
• Enter start & end of period in top then cut in half
then quarters – these are the x values
• Fill in the y values (peak, trough & neutral) where
they occur
Example: Graphing Real World Function
Use a sine function to graph a sound wave with a period of 0.004 s and
an amplitude of 3 cm.
period
amplitude
Use a horizontal scale where one unit
represents 0.004 s to complete one full
cycle. The maximum and minimum values
are given by the amplitude.
Other Transformations
• Vertical or Phase Shift – up/down or left/right
– h (left/right) and k (up/down) values in function
– Primarily focus on Vertical (Up/Down)
• Set a new neutral or Zero value
• y = sin(x – h) + k
• y = cos(x – h) + k.
• Vertical translation by k units moves the graph
up (k > 0) or down (k < 0)
Reciprocals of Sin/Cos/Tan
• Reciprocal of Sine is Cosecant = 1/Sin
– Hypotenuse over Opposite : csc
• Reciprocal of Cosine is Secant = 1/Cos
– Hypotenuse over Adjacent : sec
• Reciprocal of Tangent is Cotangent = 1/Tan
– Adjacent over Opposite : cot
Examples
• Find Csc, Sec and Cot of Θ
• Csc = 15/12 = 1.25
• Sec = 15/9 = 1.66 (or 5/3)
• Cot = 9/12 = .75
Helpful Hint
In each reciprocal pair of trigonometric functions, there is exactly one “co”
Trigonometric Identities
Algebra 2 – 11.2
• Use to compare and simplify trigonometric
functions
• Based on following table and algebraic solving
Trig Identity Examples
• : sinθcotθ = cosθ
• :
• : secθ – tanθ sinθ
• Using calculator :
– Enter into Y1 & Y2
– Compare Graphs
Inverse Trig Functions
• Going from value to angle measure
• On calculator – sin-1(a) or cos-1(a) or tan-1(a)
• Get there by 2nd SIN/COS/TAN then enter the
value in the parentheses
– Value for sin/cos must be -1≤a≤1
• Example:
– Find m<θ
: sinθ = 7/14
: sinθ = .5
: sin-1(.5) =
7
14
θ
Restrictions on Inverse Functions
• Domains & Ranges are restricted as follows: